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#### ThePerfectHacker

##### Well-known member

- Jan 26, 2012

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The meaning of $\int f(x) ~ dx$ is the collection of all functions $F(x)$ such that $F'(x) = f(x)$. It is assumed that the function $f(x) = x^{-1}$ is defined with domain on $(-\infty,0)\cup (0,\infty)$. It is true that $\log |x|$ is defined on the same domain as $x^{-1}$ and on top of that $(\log |x|)' = x^{-1}$. Thus, many people conclude (incorrectly) that therefore $\log |x| + C$ are all anti-derivatives of $x^{-1}$.

To see what this is wrong define the function,

$$ H(x) = \left\{ \begin{array}{ccc} 0 & \text{if} & x< 0 \\ 1 & \text{if} & x>0 \end{array} \right. $$

It is not hard to see that $H'(x) = 0$ and so $\log|x| + H(x)$ is an example of an anti-derivative of $x^{-1}$ which does not have the form $\log |x| + C$.

The reason why this mistake is common is because people use the following theorem: if $f(x),g(x)$ are functions defined on an open interval $I$ with $f'(x) = g'(x)$ then $f(x) = g(x) + C$ for some constant $C$. The mistake above follows from the fact that $(-\infty,0)\cup (0,\infty)$ is no longer an interval.

Whenever I teach logarithms in Calculus I always write instead,

$$ \smallint x^{-1} ~ dx = \log x + C, ~ x>0 $$

I tell my students that $x>0$ means that we are considering the function $x^{-1}$ on the interval $(0,\infty)$. With this restriction there is no longer need for absolute value and the equation is now true.

Post other calculus mistakes that are very common.